A student says that he had applied a force `F=-ksqrtx` on a particle and the particle moved in simple harmonic motion. He refuses to tell whether k is a constant or not. Assume that he has worked only with positive x and no other force acted on the particle

Options

A. As x increases k increaes
B. As x inceases k decreases
C. As x increases k remains constant
D. The motion cannot be simple harmonic

Answer

To solve the problem, we need to analyze the force applied to the particle and determine if it can lead to simple harmonic motion (SHM). The force given is \( F = -k \sqrt{x} \).

### Step 1: Understand the force equation
The force acting on the particle is given by:
\[ F = -k \sqrt{x} \]
where \( k \) is a constant (or not) and \( x \) is the position of the particle.

### Step 2: Relate force to acceleration
According to Newton's second law, the force can also be expressed as:
\[ F = m a \]
where \( m \) is the mass of the particle and \( a \) is its acceleration.

### Step 3: Express acceleration in terms of position
In simple harmonic motion, the acceleration \( a \) can be expressed as:
\[ a = -\omega^2 x \]
where \( \omega \) is the angular frequency. Therefore, the force can also be written as:
\[ F = -m \omega^2 x \]

### Step 4: Set the two expressions for force equal
Since both expressions represent the same force acting on the particle, we can set them equal to each other:
\[ -k \sqrt{x} = -m \omega^2 x \]

### Step 5: Rearrange the equation
Removing the negative signs and rearranging gives:
\[ k \sqrt{x} = m \omega^2 x \]

### Step 6: Solve for k
Dividing both sides by \( \sqrt{x} \) (assuming \( x > 0 \)):
\[ k = m \omega^2 \sqrt{x} \]

### Step 7: Analyze the relationship between k and x
From the equation \( k = m \omega^2 \sqrt{x} \), we can see that \( k \) is directly proportional to \( \sqrt{x} \). This means that as \( x \) increases, \( k \) also increases.

### Conclusion
Since we have established that \( k \) increases with \( x \), the correct option is:
- As \( x \) increases, \( k \) increases.

### Final Answer
The correct option is: **As x increases, k increases.**

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